Question

The value of \[\theta \]satisfying the given equation \[\cos \theta + \sqrt 3 \sin \theta = 2\], is
A. \[\dfrac{\pi }{3}\]
B. \[\dfrac{{5\pi }}{3}\]
C. \[\dfrac{{2\pi }}{3}\]
D. \[\dfrac{{4\pi }}{3}\]

Answer 1

Hint: The given equation will first be divided by both sides. After entering the values for the trigonometric ratios of the standard angles, use the formula for the difference of two angles to solve the remaining problem, and then determine which of the above-mentioned possibilities is accurate. Therefore, it is important to understand how to use trigonometric identities and to build identities in the correct order. It simplifies the problem and aids in your quest for the right response. In this case, \[\cos \theta + \sqrt 3 \sin \theta = 2\], using of trigonometry identities and cancelling common terms and factoring resolves to the solution.





Complete step by step solution:We have given that the equation
\[\cos \theta + \sqrt 3 \sin \theta = 2\]
Dividing the whole equation by 2 we will get:
\[ \Rightarrow \dfrac{1}{2}\cos \theta + \dfrac{1}{2}\sqrt 3 \sin \theta = \dfrac{2}{2}\]
Simplify the term on the right side:
\[ \Rightarrow \dfrac{1}{2}\cos \theta + \dfrac{1}{2}\sqrt 3 \sin \theta = 1\]
We already know that the values of,
\[\cos {60^\circ } = \dfrac{1}{2}\]; \[\sin {60^\circ } = \dfrac{{\sqrt 3 }}{2}\]
Replacing the above values in the equation we will obtain:
\[ \Rightarrow \cos {60^\circ }\cos \theta + \sin {60^\circ }\sin \theta = 1\]
Now solving the equation according to the formula\[\cos A\cos B + \sin A\sin B = \cos (A - B)\], where\[A = {60^\circ }\]and\[B = \theta \]
On simplifying, we will obtain that
\[ \Rightarrow \cos \left( {{{60}^\circ } - \theta } \right) = 1\]
After solving, it resolves to
 \[\cos {0^\circ } = 1\]
Now replace the value 1 with the obtained term\[\cos {0^\circ } = 1\]:
\[ \Rightarrow \cos \left( {{{60}^\circ } - \theta } \right) = \cos {0^\circ }\]
From the above equation, cancel the similar terms:
\[ \Rightarrow {60^\circ } - \theta = {0^\circ }\]
Hence the value of\[\theta = {60^\circ }\]



Option ‘B’ is correct

Note: Student must keep one thing in mind that\[\sin \theta \]and\[\sin \times \theta \] because it doesn’t represent a product, it represents a ratio and this is true for all the trigonometric ratios. When "n" is any integer, any trigonometric function of angle \[{\theta ^\circ }\] is equivalent to any trigonometric function of angle\[n \times {360^\circ } + \theta \]. We must commit the key formula to memory in order to solve this kind of problem, and we can only do this through practice.